Is $\lim na_n \to 0$ sufficient for $\sum a_n$ to converge? Or additional criteria is required? E.g. $a_n$ needs to be positive? Is naïve comparison with $\frac {1}{n^p}$ series justifies that ? Or is there an obvious counterexample ?
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It's insufficient. For example, if $a_n = \frac{1}{n \ln\ln n}$, then $\lim na_n = 0$, but $\sum_{n = 2}^\infty a_n$ diverges (by the comparison test). However, if $a_n$ is a sequence of positive numbers such that $\lim n^p a_n = 0$ for some $p > 1$, then $\sum_{n = 1}^\infty a_n$ converges. This is consequence of the limit comparison test. Indeed, since $\lim \frac{a_n}{1/n^p} = 0$ and $\sum_{n = 1}^\infty \frac{1}{n^p}$ converges, by the limit comparison test, the series $\sum_{n = 1}^\infty a_n$ converges.
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